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it follows that all the points on the Cartesian plane are likewise
non-demimerable. The algebraic numbers are spotted over the
plane like stars against a black sky; the dense blackness is the
firmament of the transcendentals.
The most remarkable thing about Cantor's proof is that it
provides no means whereby a single one of the transcendentals
can be constructed. To Kronecker any such proof was sheer
nonsense. Much milder instances of 'existence proofs' roused
his wrath. One of these in particular is of interest as it prophe-
sied Brouwer's objection to the full use of classical (Aristo-
telian) logic in reasoning about infinite sets.
A polynomial axn -{- bxn~'L -f- ... + Z, in which the coeiS-
cients a, b, ... I are rational numbers is said to be irreducible if
it cannot be factored into a product of two polynomials both of
which have rational number coefficients. Now, it is a meaningful
statement to most human beings to assert, as Aristotle would,
that a given polynomial either is irreducible OTIS not irreducible.
Not so for Kronecker. Until some definite process, capable
of being carried out in a finite number of non-tentative steps, is
provided whereby we can settle the reducibility of any given
polynomial, we have no logical right, according to Kroaecker,
to use the concept of irreducibility in our mathematical proofs.
To do otherwise, according to him, is to court inconsistencies in
our conclusions and, at best, the use of 'irreducibility* without
the process described can give us only a Scotch verdict of 'not
proven*. All such non-constructive reasoning is - according to
Kronecker - illegitimate.
As Cantor's reasoning in his theory of infinite classes is
largely non-constructive, Kronecker regarded it as a dangerous
type of mathematical insanity. Seeing mathematics headed for
the madhouse under Cantor's leadership, and being passion-
ately devoted to what he considered the truth of mathematics,
Kronecker attacked *the positive theory of infinity' and its
hypersensitive author vigorously and viciously with every
weapon that came to his hand, and the tragic outcome was that
not the theory of sets went to the asylum, but Cantor. Kron-
ecker's attack broke the creator of the theory.
In the spring of 1884, hi his fortieth year, Cantor experienced